1.1. What exactly are the limitations of current deep neural network models as brain models?

DNN models are essentially machine learning constructs, typically optimized for specific tasks rather than biological plausibility. Consequently, these models rely on learning mechanisms and architectural choices that are only loosely aligned with our knowledge of the brain [3–8]. A key example is supervised backpropagation, which is ubiquitous in deep learning. ‘Supervised’ means networks are provided with an externally provided ground truth. While supervised learning signals are assumed to guide some (usually motor) processes in the brain [9–11], supervised mechanisms are generally considered much less likely than unsupervised mechanisms for guiding sensory learning in the brain [12–15]. ‘Backpropagation’ refers to the updating rule used for DNN which updates model parameters hierarchically from target to input, calculating partial derivatives at each layer and propagating them back [16]. In biological networks, such a mechanism becomes increasingly unlikely the more layers are involved [3,17–21]. Despite being powerful functional models, current DNN models (especially when trained with supervised backpropagation), are not plausible models of the brain.

While past neuroconnectionist research has largely focused on architecture [22–29], more recent work has shown that training objectives play a comparably large role [30]. For example, in visual object recognition neuroscience, models have traditionally used supervised objectives and backpropagation-based optimizers [22–29]. However, unsupervised objectives like SimCLR [31] have emerged as more biologically plausible alternatives that can yield equally effective or better representations [30,32]. Nevertheless, these unsupervised rules are still often based on information-theoretical ideas rather than biological principles. A promising avenue of research is to look to neuroscience for inspiration on biologically plausible learning mechanisms for DNNs. Several such approaches have been proposed.

1. Hebbian learning rules, inspired by synaptic plasticity, are intuitively appealing but often too crude to learn complex relations [33–35].
2. Reinforcement learning aims to capture reward-based learning, but current models are not as effective as classical deep learning optimization [36–39].
3. Bayesian learning rules provide a convenient connection between probability/information theory and neuroscience. Bayesian learning [40,41] in general treats the brain as a ‘prediction machine’, by actively predicting future states and trying to optimize these predictions. The family of Bayesian learning rules includes more specified theories of neural learning, such as the free energy principle [42,43] and predictive coding [44,45]. These theoretical frameworks are more directly inspired by empirical results from neurophysiology and provide sufficient detail and operationalization to actually implement it in a computer. Due to their biological plausibility and clear cut formalization, they provide a well-suited candidate framework for building biologically plausible artificial neural networks.

In this research article, we will therefore focus on predictive coding models as the most suitable class of biologically plausible DNN models of the brain.

1.2. What is predictive coding?

Predictive coding (PC) posits that the brain is fundamentally engaged in generating predictions about incoming sensory inputs and updating internal models based on the mismatch between predictions and observations. The primary function of PC is to approximate the prediction (prior) as closely as possible to the actual future stimulus (posterior), enabling timely adaptation to new circumstances. Further, as an energy-minimizing procedure, PC proposes that only the residual, unpredicted information should be propagated to higher levels, as the expected information can be suppressed. The main computational operation is a subtraction between the prediction and the incoming stimulus (although more recent formulations also emphasize multiplicative operations such as the precision of prediction errors, linked to biological concepts such as neural gain control [46,47]). This PC framework has gained significant traction in neuroscience, as it provides a principled account for phenomena like hierarchical representations and error signals driving learning and adaptation [44–49].

The most popular PC-inspired algorithm is PredNet by Lotter et al. [50]. PredNet combines a convolutional neural network (CNN) with a long short-term memory (LSTM) recurrent layer and an autoregressive/predictive target, resulting in good performance on video prediction tasks. Recently, an alternative model - namely the Forward-Forward algorithm - has been proposed by Hinton [51]. It integrates ideas from Boltzmann machines, generative adversarial networks, and contrastive learning to form a training setting that can resemble the PC mechanism under the right conditions. Additionally, a range of other algorithms have been proposed, including work such as Millidge et al. [17,52] or Whittington & Bogacz [53]. These approaches often consist of custom architectures and custom updating rules, with errors explicitly calculated. While these approaches are inspired by biological processes, they are typically validated through proof-of-concept demonstrations showing they can learn useful representations or perform specific tasks, rather than through rigorous tests of their mechanistic biological plausibility [48]. Consequently, only a few of these models have been seriously considered as frameworks for understanding actual brain function [50,54].

1.3. What is the main goal of this work?

As interest in using DNNs to model brain processes grows, the field remains largely focused on high-performing architectures with limited biological grounding. It thus remains unclear to what extent PC-inspired DNNs realistically approximate neural dynamics or learning mechanisms in the brain. While there has been research arguing that PC-inspired networks provide various advantages to standard DNN models [55,56], such publications usually focus on theoretical or functional advantages, rather than on whether such models actually reproduce the behaviour expected from PC. To our knowledge, this paper presents the first systematic and independent evaluation of whether such biologically inspired models can serve not only as functional learning systems, but also as mechanistic models of biological neural computation. To address this, we compare several representative algorithms in a tightly controlled experimental setting. While the study does not aim to benchmark large-scale models optimized for performance, it focuses instead on understanding the mechanistic plausibility and dynamic properties of simpler, biologically motivated architectures.

This work aims to investigate whether PC inspired algorithms match phenomena or brain data better than classical supervised backpropagation in a simple perceptive setting. This would situate PC inspired algorithms as a promising alternative to other less biologically plausible approaches. To reach this goal, we compare PC inspired optimization approaches applied to one common simple Recurrent Neural Network (RNN) architecture (consisting of a feedforward input kernel and one square recurrent kernel). Since here we aim at comparing general PC-inspired algorithms and updating rules in a well-controlled setting, we focus on two candidates: A Predictive implementation, inspired by PredNet [57] and a Contrastive implementation, based on Forward-Forward [51]. We train these methods on a simple visual perception task, as a sufficiently complex and intuitively understandable task to demonstrate the effects of PC. As previous literature suggests that applying activity regularization to a network can by itself introduce PC-like effects in a network [58], we introduce an additional condition, where each of these models is trained with activity regularization applied next to the main objective. We then test whether these PC-inspired neural network models exhibit key neural signatures of predictive processing, in comparison to standard supervised and untrained neural networks. Specifically, we define three necessary landmarks which a PC network should exhibit:

1. Mismatch responses (MMR), which reflect the network’s ability to detect and respond to unexpected or deviant stimuli, and is core to the concept of saving energy in PC literature [59–61].
2. Prior expectations (i.e., actively maintaining predictions about upcoming states), which are a necessary precondition for a PC system [62–68].
3. Learning of semantic information (i.e., the association of a stimulus to an abstract encoded concept over the course of training) without explicit supervision. This is a necessary attribute of any learning system.

We use these landmarks as dependent variables, quantifying to what extent the mentioned phenomena emerge in different model types (see Fig 1). Furthermore, since gain control is considered a crucial component of PC and enables flexible context-sensitive weighting of prediction errors [46,47], we perform an additional analysis to investigate whether ANN weight regularization can be used to simulate gain control.

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Fig 1. Illustration of the different training objectives (main experimental conditions) and evaluation criteria (dependent variables).

A. Network Conditions: Depiction of each training condition (Predictive, Contrastive, Supervised), illustrating the simple Recurrent Neural Network Architecture (RNN) used as well as the optimization objective of each network. For the predictive algorithm, the supervised objective can only be applied using “global” gradient propagation (backpropagation) through the entire network. However, the Predictive and the Contrastive objective can be applied on a “local” scale, with each layer being trained on the inputs and outputs of the current layer. B. Evaluation criteria: Illustration of the three dependent variables to assess phenomenological hallmarks of PC in the networks.

https://doi.org/10.1371/journal.pcsy.0000076.g001

2.1. Mismatch responses

In PC, the occurrence of MMR is directly tied to deviations from what was expected, resulting in elevated activity [71]. We expect similar MMR patterns in PC-inspired networks. To assess the models’ ability to detect and respond to unexpected or deviant stimuli, we evaluated their MMR (measured by deviations in the evoked neural response) to a series of matching and mismatching input sequences (see Fig 2). The matching series condition consisted of a sequence of MNIST images with random augmentations, while the mismatching series condition included random augmentations and random switches in the semantic content of the images (e.g., a 3 suddenly changing to a 7). We ensured that the match and mismatch condition did not show any base level differences that might result in a difference in mean-squared activation from the network’s layers (for details see Methods 4.4.1.). We then measured the mean square layer activations of the networks over time, similar to evoked responses. Our metric of interest was whether the match and the mismatch condition produced significant deviations in the model output for the sample immediately following the stimulus switch.

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Fig 2. Example image sequences for the matching and mismatching conditions.

The top row shows a sequence of MNIST digit images with random shifts and noise, but no changes in the semantic content (matching condition). The bottom row shows a sequence with the same types of visual transformations, but with random switches in the digit identity (mismatching condition).

https://doi.org/10.1371/journal.pcsy.0000076.g002

For networks without activity regularization (see Fig 3; for details see Section D1 of S1 Appendix), most of the PC-inspired conditions showed significant MMR. Specifically, the Contrastive global (T(11998)=-41.58071, p(corrected)<1e-5), Contrastive local (T(11998)=-30.00066, p(corrected)<1e-5), and Predictive local (T(11998)=-40.36738, p(corrected)<1e-5) conditions exhibited significant changes in mean squared layer activity in response to a semantic stimulus switch. The Predictive global condition did not show a significant overall MMR (T(11998)=0.22222, p(corrected)=1). Visual inspection (see Fig C2 in S1 Appendix) revealed that this was due to a positive MMR in the first layer and a negative MMR in the second layer, which canceled each other out in the overall sum (for further discussion, see Discussion 3.1.). Neither the Supervised (T(11998)=2.28285, p(corrected)=1) nor the Untrained (T(11998)=0.72614, p(corrected)=1) conditions showed a significant MMR immediately after the stimulus switch.

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Fig 3. Mismatch responses for unregularized conditions: A. Predictive global; B. Predictive local; C: Contrastive global; D: Contrastive local, E: Supervised; F: Untrained.

Each subfigure shows the Mean Squared Average activation of a layer over different time steps, separated for sequences when an unexpected change is happening versus when an expected change is happening. Shaded areas show 95% Confidence intervals.

https://doi.org/10.1371/journal.pcsy.0000076.g003

To investigate whether these effects persist if an additional energy constraint is added, we turned to activity-regularized networks. For activity-regularized networks (see Fig 4; for details see Section D2 in S1 Appendix), all trained conditions, including Predictive global (T(11998)=-11.68170, p(corrected)<1e-5), Predictive local (T(11998)=-20.63438, p(corrected)<1e-5), Contrastive global (T(11998)=-23.97432, p(corrected)<1e-5), and Contrastive local (T(11998)=-15.89864, p(corrected)<1e-5). Even Supervised (T(11998)=17.60957, p(corrected)<1e-5), exhibited a significant MMR, however it is important to note that in this condition the effect was inverted, meaning that the mismatch condition produced less overall activity than the matching condition. The overall effects suggest that activity regularization reinforces or even produces MMR-like effects in artificial neural networks (this effect is further elaborated in Discussion 3.1.). The Untrained condition did not show significant MMR effects (T(11998)=0.72614, p(corrected)<1e-5).

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Fig 4. Mismatch responses for activity regularization condition: A. Predictive global; B. Predictive local; C: Contrastive global; D: Contrastive local, E: Supervised; F: Untrained.

Each subfigure shows the Mean Squared Average activation of a layer over different time steps, separated for sequences when an unexpected change is happening versus when an expected change is happening. Shaded areas show 95% Confidence intervals.

https://doi.org/10.1371/journal.pcsy.0000076.g004

In summary, the PC-inspired conditions, especially the locally trained ones, were generally able to generate clear mismatch responses, with the exception of the Predictive global condition. Activity regularization appeared to induce MMR-like patterns across all trained models, and even result in (inverted) MMR-like behavior in the Supervised condition.

2.2. Prior expectations

Prior predictions are a necessary component of a PC system. They allow the network to match up the prior prediction with incoming information and save energy by only propagating relevant information [62–68]. To measure to what degree the neural network models formed an inherent prior representation, we evaluated the similarities between the neural network’s prior state (negative latent state times recurrent kernel) and the expected future input. If a neural network works according to PC principles, these prior states should start to approximate the future input states over the course of learning. To investigate this effect, we introduced a series of stimuli to all networks and correlated its prior state (latent state multiplied by recurrent kernel) with the actual incoming stimulus. A high correlation in this context means that the prior state approximates the future input well, while a zero correlation means that the prior state and the future incoming input are unrelated (for details, see Methods 4.4.2.).

For unregularized networks, Predictive global (r = 0.80507) and Predictive local (r = 0.39563) showed clear correlations between the prior and the next stimulus. In contrast, Contrastive global (r = 0.01178), Contrastive local (r = -0.01456), Supervised (r = -0.00260), and Untrained (r = -0.00249) did not exhibit any prior-stimulus correlations. Accordingly, only Predictive global (Z(11998)=61.07541, p(corrected)<1e-5) and Predictive local (Z(11998)=23.05043, p(corrected)<1e-5) correlated significantly stronger than the Untrained condition. Supervised (Z(11998)= -0.00612, p(corrected)=1.), Contrastive global (Z(11998)=0.78126, p(corrected)=1.) and Contrastive local (Z(11998)=-0.66089, p(corrected)=1.) did not show prior-stimulus correlations significantly stronger than the untrained condition (see Fig 5; for details see Section D1 in S1 Appendix).

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Fig 5. Correlations between priors (project to input level) and future input.

Left: No regularization, Right: With activity regularization. High correlation means that the prior (negative recurrent state) projected back to the input level correlates highly with the next (unseen) stimulus in the sequence. Low correlation means the state and the next stimulus are uncorrelated. Solid line: p < 0.0001.

https://doi.org/10.1371/journal.pcsy.0000076.g005

The same pattern held true for the activity-regularized networks: Predictive global (r = 0.78028) and Predictive local (r = 0.57463) demonstrated strong prior-stimulus correlations. Meanwhile, Contrastive global (r = 0.00826), Contrastive local (r = -0.00059), Supervised (r = -0.01706), and Untrained (r = -0.00249) did not show any clear prior formation. Like in the unregularized condition, only Predictive global (Z(11998)=57.41855, p(corrected)<1e-5) and Predictive local (Z(11998)=35.97105, p(corrected)<1e-5) correlated significantly stronger than the Untrained condition. Supervised (Z(11998)= -0.79785, p(corrected)=1.), Contrastive global (Z(11998)=0.58876, p(corrected)=1.) and Contrastive local (Z(11998)=0.10407, p(corrected)=1.) did not show prior-stimulus correlations significantly stronger than the untrained condition (see Fig 5; for details see Section D2 in S1 Appendix).

In summary, the Predictive condition was the only one that consistently exhibited the formation of meaningful prior expectations across the different network configurations. The other conditions, including Contrastive, Supervised, and Untrained, did not show evidence of prior formation.

2.3. Learned representations

To assess each model’s ability to learn abstract representations, we examined the encoded information content in each model by decoding the original number classes represented by the input from the model’s output state. While the Supervised condition has been explicitly trained to fulfil this task, the PC conditions were optimized on unrelated unsupervised targets. Accordingly, we use better-than-untrained decoding performance as an indicator of the PC network’s capability of learning semantic information under unsupervised learning conditions.

The Predictive global condition (accuracy = 0.38667, SE = 0.01232) not only performed notably better than the untrained network (T(11998)=7.57907, p(corrected)<1e-5), but not significantly better (T(11998)=2.26123, p(corrected)=0.35644) than the Supervised condition (accuracy = 0.36667, SE = 0.01219), which still performed notably better than Untrained (T(11998)=5.31126, p(corrected)<1e-5). The Contrastive global condition (accuracy = 0.36550, SE = 0.01219) also showed substantial learning beyond the Untrained condition (T(11998)=5.17835, p(corrected)<1e-5), though insignificantly lower than Predictive global (T(11998)=2.39393, p(corrected)=0.25026). The Contrastive local condition (accuracy = 0.35217, SE = 0.01209) (T(11998)=3.65364, p(corrected)=0.00389) as well as the Predictive local condition (accuracy = 0.34700, SE = 0.01204) (T(11998)= 3.05970, p(corrected)=0.03331) still exhibited significant learning above the Untrained baseline (accuracy = 0.32000, SE = 0.01180). These results are illustrated in Fig 6 (For details see Section D1 in S1 Appendix).

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Fig 6. Learning Performance evaluated by classification accuracy of the original stimulus classes based on the output encoding of the neural network models.

Left: No regularization, Right: With activity regularization. Dotted line: p < 0.05. Solid line: p < 0.0001.

https://doi.org/10.1371/journal.pcsy.0000076.g006

Combining activity regularization with the predictive algorithm resulted in a notable decline of learning performance (see Discussion 3.1.). In this condition, the learning effects diverged more across between models: Predictive global (accuracy = 0.33300, SE = 0.01192) and Predictive local (accuracy = 0.23017, SE = 0.01065) both showed decreased learning performance, with Predictive global performing worse than Supervised (T(11998)=2.82368, p(corrected)=0.07133) and on par with an Untrained Network (T(11998)=1.42093, p(corrected)=1.) Predictive local even performing notably worse than the Untrained condition (T(11998)=-12.62837, p(corrected)<1e-5). In contrast, Contrastive global (accuracy = 0.36600, SE = 0.01219)(T(11998)=5.23532, p(corrected)<1e-5) and Contrastive local (accuracy = 0.37350, SE = 0.01224)(T(11998)=6.08825, p(corrected)<1e-5) exhibited increased learning compared with the Untrained condition. The Supervised condition still performed notably better than the untrained condition (T(11998)=4.24578, p(corrected)=0.00033). These results are illustrated in Fig 6 (For details see Section D2 in S1 Appendix).

To conclude, our analysis of semantic learning effects showed that all algorithms in almost all settings exhibited significant learning of category information beyond the untrained baseline condition. In the unregularized setting, all training conditions showed significant learning effects beyond the untrained baseline. While this is to be expected for the Supervised condition, it shows that the PC objectives as well are capable of instilling semantic information into a model despite their unsupervised training objective.

2.4. Generalization to real video data

To assess whether the main findings generalize to more complex, real-world data, we repeated the full analysis on the BAIR Robot Pushing dataset. This dataset introduces greater visual and semantic complexity, as well as continuous target variables. The results confirmed our main findings: both Predictive conditions exhibited strong prior formation, with high correlations between the internal prior and the upcoming stimulus. Significant mismatch responses were also observed in the Predictive local and both Contrastive conditions, mirroring the main analysis. While the learning evaluation showed similar trends—with Predictive and Supervised conditions outperforming Contrastive and Untrained models—the differences were not statistically significant under multiple comparison correction. Nonetheless, the pattern of results closely replicates the structure observed in the original analysis, indicating that key effects of prior formation and mismatch sensitivity robustly extend to more complex visual input and continuous output spaces. Overall, this supplementary analysis supports the generalizability of our core conclusions beyond the original experimental setting. The full replication is reported in Section A in S1 Appendix.

2.5. Gain control

In PC, the noise level of the output signal is correlated with the noise level of the input. Gain control is the ability to manipulate the output noise level independently of the input noise [46,47]. As gain control is an important mechanism in PC literature [46], we investigated whether it is possible to simulate this mechanism in PC networks using weight regularization. We created two input conditions with high and low noise levels and evaluated whether weight regularized networks significantly reduce the difference in noise level of the model’s output. We did this by comparing the variance ratio between low and high noise input for unregularized versus weight regularized networks.

Our results showed that weight regularization results in a gain control-like effect, reducing the neural activity’s variance ratio between high noise and low noise input stimuli. Contrastive (Z = -67.08483, p(corrected)<1e-5), Predictive (Z = -64.67718, p(corrected)<1e-5), and Supervised (Z = -67.08477, p(corrected)<1e-5) all showed significant reduction in output variance ratios between high noise and low noise when weight regularized. From visual inspection of Fig 7, it can be seen that all algorithms have undergone notable reduction in variance differences with applied weight regularization. However, weight regularization did not lead to perfectly identical output variance distributions for any algorithm, with Contrastive (Z = -67.08478, p(corrected)<1e-5), Predictive (Z = -66.92948, p(corrected)<1e-5), and Supervised (Z = -36.19967, p(corrected)<1e-5) all still showing significance variance difference between high noise and low noise samples (for details see Section D4 in S1 Appendix).

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Fig 7. Distributions of output activations for unregularized and weight-regularized networks, for high noise and low noise inputs.

Output activations from high noise inputs are shown in blue, output activations from low noise inputs are shown in orange. Weight regularization leads not only to normalization of the neural output distribution, but also reduces the ratio in output variance in response to high noise and low noise inputs.

https://doi.org/10.1371/journal.pcsy.0000076.g007

3.1. To what extent did the PC inspired models exhibit landmarks of biological PC?

Mismatch Responses: All PC-inspired networks showed some form of MMR (i.e., stimulus mismatch induced positive activity) in at least one layer. We found the strongest MMR in the Contrastive condition, likely due to the fact that the Forward-Forward implemented here is directly optimized towards maximizing the output activity after occurrence of mismatching stimuli. The Predictive local condition also evoked clear MMR. Interestingly, the Predictive global condition produced a negative MMR in the first layer and a positive MMR in the second layer, which cancelled out when added together (see Fig C2 in S1 Appendix). This phenomenon likely arises from the fact that the globally defined predictive function is defined to facilitate MMR at the final layer, which might lead to opposite effects in the previous layers, meaning that (goal-oriented but negative) weight patterns that are punished in the final layer are more likely to appear in previous layers. A comparable pattern is observed in the brain as well, where repetition suppression and repetition enhancement can occur at different levels of the cortex [47]. However, further research is needed to determine if it is analogous to neurophysiological MMR or an unrelated occurrence in the artificial network. Overall, the clear MMR spikes in PC-inspired networks sets them apart from Supervised networks, which does not show such clear spikes. These results provide further evidence that MMR are a consequence of prediction-based learning, as claimed in PC literature [59]. They further confirm that complex semantic information can be learned in an unsupervised fashion by maximizing the contrast in MMR (as proposed in [51]).

MMR responses were strongly enhanced or even evoked by activity regularization. This suggests activity regularization may be a promising approximation of the energy-saving principle behind PC [58]. Interestingly, we find that activity regularization evokes an inverted MMR-like effect in Supervised recurrent networks, where the output activity for mismatching stimuli is lower than for matching stimuli. This does not directly align with the idea of PC, where energy should be saved by only propagating unexpected information, meaning that mismatch activity is expected to be larger than expected activity. This effect can be explained by the combination of the Supervised learning objective in combination with activity regularization: Supervised RNN models have the tendency to build up activity over time through accumulating activation over the recurrent state (this is an issue related to the exploding gradients problem and discussed in [72,73]). In unregularized networks, this leads to an exponential increase in activity (see Fig 3) as long as the network is presented with unchanching patterns that accumulate and amplify in the recurrent state. Now when a mismatching input appears, the activity builds up slightly slower because the new input tends to cancel out activity rather than further amplifying existing patterns. In the activity regularization setting, this activity accumulation is decreased due to increased penalization of high activity (see Fig 4). However, similar to the unregularized condition, the stimulus switch tends to cancel out activity rather than amplifying existing patterns, leading to reduced activity for the mismatch condition. This might explain the effect of the MMR-like pattern in the Supervised model with activation regularization.

Further, we found that the application of activity regularization in combination with a Supervised objective, resulted in unexpected behavior. The Predictive condition performed well overall, but was heavily impaired when combined with activity regularization. An interesting aspect of this phenomenon is that the Predictive models with activity regularization still exhibited very strong correlations between the current prior and the future state, but seemed to encode drastically less class-specific semantic information. This suggests that the activity regularization forced the loss in the Predictive objective to only focus on the Predictive information which is shared between classes, creating relatively high predictions from this generally shared variance, while at the same time cutting all the activity that would carry important information for specific classes, but which occur less often and therefore contribute too little signal to withstand the activity regularization.

Prior expectations: Contrary to the broad occurrence of MMR, only the Predictive condition showed clear prior formation, while the Contrastive, Supervised, and Untrained conditions did not. Empirical evidence shows that predictions in the brain are usually organized topographically. This has been widely shown in the visual, auditory, sensory, and other systems [74–79]. As PC builds upon this idea of neuron-by-neuron topographically ordered predictions, this suggests that a population-activity based contrastive model such as Forward-Forward [51] might not be a perfectly suitable model of PC. Further, this suggests that explicit definition of a prior in the Predictive algorithm may lead to the appearance of an MMR as a secondary effect, while the opposite (i.e., the explicit definition of a MMR in the Contrastive algorithm implying the formation of a topographically organized prediction) does not hold. This insight can be further used to investigate alternative models of PC and Bayesian brain theories [48,49,59,80–83]. MMR are among the main phenomena that lead to the development of the theory of PC and are frequently used as evidence for it [47,59]. If there are learning systems that exhibit MMR, but do not exhibit explicit prior predictions, then models such as the ones presented here can help in investigating the theory by evaluating whether a predictive MMR model or a non-predictive MMR model fits the brain better.

Learned Representations: Finally, to assess the learning effects of the different training approaches, we compared the performance of the models on the task of classifying the original MNIST digit classes based on the encoded representations in the networks. In general, predictive and contrastive models showed better-than-random learning effects. Supervised showed the strongest learning effects, because the algorithm was specifically optimized to perform a classification task on a similar set of input stimuli. One important aspect to consider is that the Predictive model used in this study performed well at encoding semantic information about stimuli, while similar investigation into the classical PredNet architecture [84] did not show strong encoding capabilities. This is possibly related to the lack of semantic encoding occurring in the convolution-based PredNet architecture, which has been fixed in the presented predictive model (see Methods 4.1.). While the Supervised algorithm still showed the best learning performance, our results show that PC-inspired algorithms might be useful for many learning contexts, adding to other research confirming PC’s usefulness for tasks like transfer learning and generalization [85].

Gain Control: In an additional analysis we found that weight regularization added to neural networks during the training process results in a variance normalization of the output activations. In unregularized networks, the variance of the network’s output activity correlates with the variance of the input. However, when weight regularization is applied, this correlation is weakened, meaning that noisy inputs almost produce the same output activity variance as clean inputs. We successfully showed that weight regularization can be used to implement gain control mechanisms in DNN models of the brain. Such a gain control mechanism may achieve two aims, consistent with the PC framework: 1. scaling output variance to compensate for input variance, and 2. encouraging the network to learn sparse representations and minimize activity to conserve energy [43,45,86]. Additionally, this effect should be taken into account when using DNN as neuroconnectionist models while changing the noise levels in the input (such as [87,88]). In such cases, networks trained using regularization could lead to different distributional properties of the activation, which might affect outcome or decodability in downstream (encoding, decoding, RSA) analyses.

Summary: Taken together, these results indicate that under certain circumstances, the neural network models trained with PC-inspired objectives were able to display hallmarks of predictive processing. Specifically, the locally trained predictive objective (without activity regularization) in a RNN exhibited these traits most closely. This model was able to fulfill the expected landmarks of PC, including the formation of meaningful priors, the generation of mismatch responses, and the learning of informative representations, without requiring extensive gradient propagation throughout the network hierarchy. This strongly supports the idea that simple predictive objectives can serve as valid approximations of PC. These models can exhibit behaviors resembling information processing in the brain better than standard DNN models. Our results not only show that PC is effective as a learning mechanism (which has been demonstrated in other studies; [17,50–53,89–92]), but also connect the learning effects to the phenomenological markers theorized as part of PC theory (such as priors and MMR), confirming them as a possible and logically coherent consequence of the PC mechanism.

3.2. What are the key limitations of the current study and how can they be addressed?

While this study provides valuable insights into the relationship between artificial and biological learning, it highlights the need for further exploration and refinement of PC-inspired neural network models. Future research should scale up the model complexity by exploring more sophisticated architectures with additional layers and increased capacity. This could uncover new emergent properties and better align the artificial networks with the brain’s hierarchical structure. Specifically, incorporating deeper networks with longer propagation of backwards errors could better mimic the hierarchical information processing theorized by PC.

Another very common way of showing alignment between DNN models and the brain are encoding models and benchmarks like Brain-Score [1], where models are compared in terms of how well they predict neurophysiological or behavioral data. Over recent years, there has been much progress using functionally performant DNN models like CNN or Transformers as models to investigate brain semantics and representations. While these functionally performant models are very successful at brain encoding or representational similarity tasks (i.e., explaining representational variance in neurophysiological data of humans performing semantic tasks) [1], using them to investigate specific mechanisms of neural functions is inherently limited by their mechanistic capability to adequately represent the corresponding mechanism in the human brain. For example the investigation of feedback mechanisms in the brain requires a DNN model containing manipulable variables that mechanistically represent such a feedback mechanism. In this research, we chose to investigate PC networks specifically, as a potential future model category allowing researchers to investigate variables which cannot be investigated using less biologically plausible DNN models. However, PC networks are rarely investigated in respect of semantic or representational likeness to the brain, largely due to being mostly implemented at a very small scale and therefore often performing worse than larger DNN models in representational aspects [48]. Therefore, as a starting point, we decided to focus on the phenomenological and dynamical behavior of PC networks as this mechanistic and biological interpretability is the main advantage of PC networks compared to larger DNN models. Nevertheless, it would be helpful to address semantic and representational capabilities of these networks in further research.

Another possible improvement is to test these algorithms on more challenging and diverse tasks beyond the moving MNIST and BAIR robot pushing datasets used in this work. The PredNet model [57], for example, has been evaluated on high level visual data including more complex and more diverse naturalistic inputs [93]. However, PredNet has been criticized for issues related to its biological plausibility [84]. The predictive model used in the current study aims to address these biological plausibility concerns, but it has not yet been tested on larger-scale tasks. Similarly, the contrastive model, which is based on the Forward-Forward algorithm [51], as well as other related algorithms [91], have only been evaluated on MNIST or moving MNIST, which are relatively simplified tasks. MNIST or moving MNIST may not generalize well to more complex computer vision tasks [94]. While we show that the effects presented in this publication do indeed largely generalize to real life video data, further evaluation on larger and more complex datasets which include more naturalistic conditions, background, and object interactions. By testing these PC-inspired models on a wider range of tasks, including those with more naturalistic and diverse inputs, researchers can better assess their generalization capabilities and further explore the computational principles underlying biological neural information processing.

3.3. Implications for biological and artificial neural learning

The insights from this research can be used to create better DNN models of the brain. The neuroconnectionist framework relies on creating neural network models and experimentally altering different mechanisms to investigate how changes in the model improve or decrease its similarity to the brain [1,22]. However, research into PC often relies on biologically implausible machine learning networks [26,84,95] despite previous research showing that well-performing machine learning models are not always better neuroconnectionist brain models [1,30]. The results presented here suggest that the PC-inspired models, while currently only sparsely being used as neuroconnectionist research models [50], might be a promising future direction in brain modelling.

From a machine learning perspective, the results from this paper support existing evidence that PC-inspired algorithms based on Predictive as well as Contrastive objectives [51,57] are promising methods for building effective learning algorithms. The models presented in this work perform well in their role as unsupervised machine learning models, even after addressing many of the biologically implausible aspects that have been criticized in classical deep neural networks. Specifically, such models could serve as a foundation for developing new artificial neural network models for time-dissolved or sequential tasks such as time series or video prediction. While PC-inspired models currently may not surpass existing state-of-the-art models in unsupervised learning [96–98], the elegant way in which PC integrates a prediction-based framework with complex semantic learning systems poses a strong foundation for creating simple and effective theory-driven models for unsupervised learning.

4.1. General model architecture

To implement the PC-inspired neural learning algorithms, and to facilitate comparisons between distinct models, we used a simple RNN architecture as a basis for all models (see Fig 8). We chose this straightforward RNN architecture to demonstrate the core properties of PC in a maximally simple and well-controlled setting. Specifically, all conditions were trained on a model with two vanilla RNN layers - both composed of a forward kernel as well as a recurrent kernel. The recurrent connections in the RNN allow the model to generate some form of prior expectations, which can then be integrated with incoming sensory input. Specific details on the architecture and hyperparameters are given in Section B in S1 Appendix. All code to replicate this study are published under github.com/DiGyt/predictive_coding_algorithms. Further data and pretrained models to directly replicate the results are published under https://doi.org/10.5281/zenodo.17228391.

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Fig 8. A. Illustration of the PredNet Architecture [57].

B. Illustration of the Recurrent version of the Forward Forward architecture [51]. C. Illustration of the Simple RNN Architecture used here. While PredNet is a high level approach for self-supervised video learning, we use SimpleRNN to create the most minimalistic and assumption-free type of network capable of expressing prior predictions. There are 3 main differences between the two models: (1) The top-down connections presented in PredNet are omitted in the SimpleRNN for the sake of simplicity. (2) The layers in PredNet are specified for learning of visual information, using convolutional layers, convolutional LSTMs, max-pooling layers. In the Simple RNN, we use only a minimal amount of weights, consisting of one input kernel and one recurrent kernel. (3) PredNet maintains the pixel-level state of the image throughout the entire network hierarchy, meaning that the network does not encode the image sequence, but only applies changes to it. This means that PredNet output states do not provide a biologically plausible representation of neurons that might be interpreted as pure positive firing patterns of neurons. This maintenance of the pixel-level variable in combination with using ReLU activation functions (which are only capable of expressing positive values) also requires an explicit handling and splitting up of positive and negative correction errors. In the SimpleRNN model, the states are encoded and maintained as purely positive values (interpretable as firing patterns). In this network, the expression of positive and negative error corrections takes place through positive and negative weights in the input and recurrent kernel, which are applied before the activation function is applied.

https://doi.org/10.1371/journal.pcsy.0000076.g008

By using a simple RNN rather than more complex architectures, we aimed to isolate the key PC principles without the confounding factors that could arise in deeper or more specialized network designs [99]. Further, this constant architecture allows us to isolate training effects induced by the training objective and optimization procedure, omitting effects created by varying architectures. Further details and an overview table on architectural hyperparameters for the main analysis as well as the BAIR replication analysis are reported in Section B in S1 Appendix.

4.2. Training environment

The neural network models in this study were trained on a moving MNIST [69] paradigm, which was specifically designed to present the models with a challenging yet controlled set of input stimuli. This paradigm involves a series of MNIST digit images that move across the input field in a particular order, with the digits moving at different speeds (comparable to [91]) (see Fig 9). Specifically, the movement of the MNIST digits follows sine wave patterns, with the phase and direction of the sine waves on the horizontal and vertical axes determined by the class of the number being presented. This approach encourages the networks to develop robust representations that link the digit semantics to their dynamic visual features, rather than just memorizing static pixel-level details. We use MNIST because it is a typical dataset to test new DNN architectures on, and has been used as a test dataset for various novel DNN architectures [51,91]. However, as a dataset for semantic classification, MNIST is notoriously easy to solve [100]. In order to enhance classification difficulty and improve generalization, we applied a variety of random augmentations to the input images. These augmentations included shifts (up to ~±21% of the receptive field with a standard deviation of ~14% of the receptive field) and the addition of Gaussian noise (+33% standard deviation relative to noise-free images). By introducing these transformations, we ensured that the networks could not simply memorize the pixel-level details of the inputs, but had to learn more abstract representations. Additionally, all input images were normalized to a 0–1 scale, rather than the original 0–255 range, to help standardize the data.

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Fig 9. Illustration of the input series.

Networks were trained according to a moving MNIST paradigm. For each batch, multiple samples were fed to the network. Each sample contained a series of images, depicting a number moving across the visual field. Each separate number class moves across the visual field according to specific rules (direction and magnitude in change of x and y coordinates per time step). These rules were shared between samples of the same number class, but differed from samples of other number classes.

https://doi.org/10.1371/journal.pcsy.0000076.g009

The main dataset consisted of 54,000 MNIST samples drawn from the original training set, with an additional 6,000 images held out for testing. During training, the data was shuffled each epoch and then batched up in batches of 512 samples per batch (and 240 samples for the final batch). The preprocessing for the BAIR robot pushing dataset which was used for our confirmatory analysis are reported in Section A in S1 Appendix. Further details and an overview table on training hyperparameters for the main analysis as well as the BAIR replication analysis are reported in Section B in S1 Appendix.

4.3. Model variations

4.4. Model evaluation

We selected three specific evaluation metrics - mismatch responses, prior expectations, and learned representations - to align with the key behaviors we would expect from a neural network model exhibiting the principles of PC. In the following, we describe how each of these metrics are quantified.

4.5. Analysis of gain control

We additionally investigated whether weight regularization can be used to simulate gain control. In PC, the noise level of the output signal is correlated with the noise level of the input. Gain control is the ability to manipulate the output noise level independently of the input noise [46,47]. Investigating gain control in our PC-inspired DNNs can provide insights into corresponding brain mechanisms. Weight regularization in neural networks is a widely known and used mechanic that constrains the magnitude of a network’s weights [117]. From preliminary investigations, we expected this to be a suitable mechanism to implement gain control, as weight regularization reduces the network’s sensitivity to variance in the input, potentially normalizing the statistical attributes of the output.

For this analysis, we trained all of the algorithms according to the standard setting without activity regularization, but with weight regularization. We compared the unregularized networks to the weight regularization networks. For weight regularized and unregularized networks of all algorithms, we passed single stimulus images while adding high (noise-to-signal standard deviation ratio ~1.7) or low (noise-to-signal standard deviation ratio ~0.3) Gaussian noise to the networks and collected the output activity. In order to create a distribution of variance ratios, we calculated the variances of the output activity and divided the variance of the high noise input images through the variances of the low noise input images for each sample. To evaluate whether the ratio of variance for both high and low noise images is significantly smaller for weight regularized networks than for unregularized networks (indicating gain control), we performed a Wilcoxon signed-rank test. We chose to use a nonparametric test for this analysis as the distribution of variance ratios was skewed and the requirement of normality was not given. Further we investigated whether despite weight regularization - there are still differences in variance distributions between low and high noise stimuli. We did this by evaluating if the variance distribution in the weight regularized conditions are indistinguishable between low noise and high noise stimuli. For this, we used another Wilcoxon signed-rank test. Significance levels were corrected for 6 multiple comparisons (2 tests with 3 comparisons) using a Bonferroni correction.

4.6. Other computational settings

This study was implemented in Python, using tensorflow [118], keras [119], scikit-learn [120], and scipy [121]. Other details on hyperparameters and software used are mentioned in Section B in S1 Appendix.